can be arbitrarily small among domains \( of a given diameter. Investigation of this result for \) < 0\( was motivated by the following conjecture of Andrews, Clutterbuck, and Hauer \cite [§10]{ACH} for \) 0\(:\begin{conj}[Robin gap]If \alpha \ge 0 and \mathbb {R}^n is a bounded convex domain then(\lambda _2 - \lambda _1)( \ge (\lambda _2 - \lambda _1)(I),where I \subset \mathbb {R} is an open interval of length equal to the diameter of .\end{conj}\) The Robin gap conjecture is known to hold in the Neumann case (\(\alpha = 0\) ) due to Payne and Weinberger [1]}. A new proof was given by Andrews and Clutterbuck [2]}. In the Dirichlet case (\(\alpha = \infty \) ) it is known, not only for the Laplacian, but also for Schrödinger operators with a convex potential, thanks to Andrews and Clutterbuck [2]}.
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